Download 6-4 Study Guide and Intervention Rectangles

NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-4 Study Guide and Intervention
Rectangles
Properties of Rectangles A rectangle is a quadrilateral with four right angles.
Here are the properties of rectangles. A rectangle has all the properties of a parallelogram.
• Opposite sides are parallel.
• Opposite angles are congruent.
• Opposite sides are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.
Also:
• All four angles are right angles.
∠UTS, ∠TSR, ∠SRU, and ∠RUT are right angles.
• The diagonals are congruent.
≅
Example 1: Quadrilateral RUTS above is a
Example 2: Quadrilateral RUTS above is a
rectangle. If US = 6x + 3 and RT = 7x – 2, find x.
rectangle. If m∠STR = 8x + 3 and m∠UTR = 16x – 9,
find m∠STR.
The diagonals of a rectangle are congruent, so US = RT.
6x + 3 = 7x – 2
3=x–2
5=x
∠UTS is a right angle, so m∠STR + m∠UTR = 90.
8x + 3 + 16x – 9 = 90
24x – 6 = 90
24x = 96
x=4
m∠STR = 8x + 3 = 8(4) + 3 or 35
Exercises
Quadrilateral ABCD is a rectangle.
1. If AE = 36 and CE = 2x – 4, find x.
2. If BE = 6y + 2 and CE = 4y + 6, find y.
3. If BC = 24 and AD = 5y – 1, find y.
4. If m∠BEA = 62, find m∠BAC.
5. If m∠AED = 12x and m∠BEC = 10x + 20, find m∠AED.
6. If BD = 8y – 4 and AC = 7y + 3, find BD.
7. If m∠BEA = 62, find m∠DAC.
8. If m∠DBC = 12x + 2 and m∠ABD = 10x, find m∠ABD.
Chapter 6
23
Glencoe Geometry
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-5 Study Guide and Intervention
Rhombi and Squares
Properties of Rhombi and Squares A rhombus is a quadrilateral with four congruent
sides. Opposite sides are congruent, so a rhombus is also a parallelogram and has all of the
properties of a parallelogram. Rhombi also have the following properties.
The diagonals are perpendicular.
⊥
Each diagonal bisects a pair of opposite angles.
bisects ∠RMO and ∠RHO.
bisects ∠MRH and ∠MOH.
A square is a parallelogram with four congruent sides and four congruent angles.
A square is both a rectangle and a rhombus; therefore, all properties of parallelograms,
rectangles, and
rhombi apply to squares.
Example: In rhombus ABCD, m∠BAC = 32. Find the measure of each numbered angle.
ABCD is a rhombus, so the diagonals are perpendicular and △ABE is a right triangle.
Thus m∠4 = 90 and m∠1 = 90 – 32 or 58. The diagonals in a rhombus bisect the vertex
angles, so m∠1 = m∠2. Thus, m∠2 = 58.
A rhombus is a parallelogram, so the opposite sides are parallel. ∠BAC and ∠3 are
alternate interior angles for parallel lines, so m∠3 = 32.
Exercises
Quadrilateral ABCD is a rhombus. Find each value or measure.
1. If m∠ABD = 60, find m∠BDC.
2. If AE = 8, find AC.
3. If AB = 26 and BD = 20, find AE
(Use the Pythagorean Theorem.).
4. Find m∠CEB.
5. If m∠CBD = 58, find m∠ACB.
6. If AE = 3x – 1 and AC = 16, find x.
7. If m∠CDB = 6y and m∠ACB = 2y + 10, find y.
8. If AD = 2x + 4 and CD = 4x – 4, find x.
Chapter 6
24
Glencoe Geometry